The natural spherical projection associated with the Hierarchical Equal-Area and Isolatitude Pixelization (HEALPix) is described and shown to be one of a hybrid class that combines the cylindrical equal-area and Collignon projections, not previously documented in the cartographic literature. Projection equations are derived for the class in general and are used to investigate its properties. It is shown that the HEALPix projection suggests a simple method of (i) storing and (ii) visualizing data sampled on the grid of the HEALPix pixelization, and also suggests an extension of the pixelization that is better suited for these purposes. Potentially useful properties of other members of the class are described, and new triangular and hexagonal pixelizations are constructed from them. Finally, the standard formalism is defined for representing the celestial coordinate system for any member of the class in the FITS data format.